The circle $x^2+y^2-8x-12y+\alpha=0$ lies in the first quadrant without touching the coordinate axes. If $(6, 6)$ is an interior point to the circle,then

Find the equation of the circle passing through the origin and whose center is the intersection point of the lines $2x - 3y + 4 = 0$ and $3x + 4y - 5 = 0$.

The lengths of the intercepts made by a circle $S$ on $X$ and $Y$-axes are $\frac{2 \sqrt{13}}{3}$ and $\frac{2 \sqrt{22}}{3}$ respectively. If the radius of the circle $S$ is $\frac{\sqrt{38}}{3}$ and its centre $C$ lies in the second quadrant,then $C=$

Let the equation $x^{2}+y^{2}+px+(1-p)y+5=0$ represent circles of varying radius $r \in (0, 5]$. Then the number of elements in the set $S = \{q : q = p^{2} \text{ and } q \text{ is an integer}\}$ is ..... .

Find the centre and radius of the circle $2x^{2} + 2y^{2} - x = 0$.

Let the abscissae of the two points $P$ and $Q$ be the roots of $2x^{2}-rx+p=0$ and the ordinates of $P$ and $Q$ be the roots of $y^{2}-sy-q=0$. If the equation of the circle described on $PQ$ as diameter is $2(x^{2}+y^{2})-11x-14y-22=0$,then $2r+s-2q+p$ is equal to